Fluid mechanical investigations of ciliary propulsion

OF CILIARY PROPULSION

Thesis by

Stuart Ronald Keller

In Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy

California Institute of Technology Pasadena, California

1975

ACKNOWLEDGMENTS

It is with heart-felt emotion that I express my sincere gratitude to Professor Theodore Y. T. Wu. His personal warmth and estimable knowledge have been an inspiration to my personal and professional development.

Thanks are also due to Professor Milton Plesset and Dr. Christopher Brennen for some interesting discussions and thoughtful

suggestions. I would also like to thank Dr. Howard Winet and Dr. Anthony T. W. Cheung for many stimulating conversations and for their invaluable advice and assistance on the biological portions of this work.

For their skillful aid and helpful recommendations on the preparation of this manuscript, I would like to express my sincere appreciation to Helen Burrus and Cecilia Lin.

ABSTRACT

Fluid mechanical investigations of ciliary propulsion are

carried out from t<uo points of view. In Part I, using a planar geome-try, a rnodel is developed for the fluid flow created by an array of metach.ronally coordinated cilia. The central concept of this model i::: to replace the discrete forces of the cilia ensemble by an equivalent continuum distribution of an unsteady body force within the cilia layer. This approach facilitates the calculation for the case of finite amplitude movement of cilia and takes into account the oscillatory component of

the flow. Expressions for the flow velocity, pressure, and the energy expended by a cilium are obtained for small oscillatory Reynolds num-bers. Calculations are carried out with the data obtained for the two ciliate!: Opalina ranarurn and Paramecium multimicronucleatum. The results are compared with those of previous theoretical models and some experimental observations.

In Part II a model is developed for representing the mechanism of propulsion of a finite ciliated micro-organism having a prolate

spheroidal shape. The basic concept of the model is to replace the micro-organism by a prolate spheroidal control surface at which

certain boundary conditions on the fluid velocity are prescribed. These boundary conditions, which admit specific tangential and normal

PART I

PART II·

TABLE OF CONTENTS

A TRACTION-LAYER MODEL FOR CILIARY PROPULSION

1. Introduction

2. Description of the model

3. Solution of the fluid mechanical problem 4. Force exerted by a cilium

5. Determination of the body force field

6.

Results for Paramecium and Opalina 7. Energy expenditure by a cilium 8. Conclusions and discussion

Appendix A Appendix B References

A POROUS PROLATE SPHEROIDAL MODEL FOR CILIATED MICRO -ORGANISMS

PAGE 1

2

8

13 18 21 24

36

40 42

46

49 52

1. Introduction 53

2. Geometry and boundary conditions 58

3. Solution 61

4. Self -propulsion 65

5. Energy dissipation 68

6.

The effect of shape upon energy expenditure 70

7. The stream function 7 3

9.

Concluding remarks Appendix

References Figures

PART I

1. Introduction

Cilia are found in practically all the phyla of nature. Many micro -organisms are covered with cilia, the movement of which

results in their vital locomotion. The palate of the frog is lined with cilia where their function is to maintain a constant cleansing flow of fluid. In man cilia are involved in the flow of mucous in the upper respiratory tract, the transport of gametes in the oviduct, and the circulation of the cerebrospinal fluid in which the brain is bathed.

A typical cilium is about 10 microns in length and has a circu-lar cross section having a diameter of 0. 2 microns. Although this investigation will not be concerned with the biophysics of how cilia move, it is worth noting that the internal structure of cilia has been found to be the same in many different organisms (Satir, 1974). In most ciliated systems, each cilium executes a periodic beat pattern which is indicated in Figure la. The numbers indicate successive positions of the cilium with a constant time interval between positions. The portion of the ciliary beat cycle during which the cilium is nearly straight and moving rapidly (positions 1-4) is referred to as the effec-tive stroke, while the remainder of the cycle is termed the return

stroke. While early observers (e. g. Gray, 1928) thought that all cilia execute planar beat patterns, it has recently been demonstrated that in various cases the cilia perform a three-dimensional beating motion

(Cheung, 197 3). For these cases it is often difficult to differentiate between an effective and a return stroke.

Laeoplectic

~--3

L---5

---:4

Antiplectic

t

I

Effective

Stroke

'

l

Symplectic

(b)

--~

Dexioplectic

[image:9.543.43.496.39.681.2]

the same rate, the cilia do not all beat in the same phase. The

move-ments of adjacent cilia along the same row are exactly in phase, and

these rows are each said to display synchrony. Neighboring cilia along

any other direction are slightly out-of-phase and show metachrony.

The cilia in files at right angles to the lines of synchrony show a

maxi-mum phase difference between neighbors, and waves of activity,

metachronal waves, pass along the surface in these directions. The

relationship between the direction of propagation of the metachronal

waves and the direction of the effective stroke of the ciliary beat varies

in different ciliated systems. The four basic types of metachrony were

recognized and named by Knight-Jones (1954) as shown in Figure lb. For symplectic metachrony the wave propagates in the same direction

as the effective stroke, while for antiplectic metachrony the wave

moves in the opposite direction. Laeoplectic and dextoplectic

meta-chrony correspond to the situation where the wave moves in a direction

perpendicular to that of the effective stroke. This nomenclature is

inadequate, of course, for cilia not displaying a clearly defined

effec-tive stroke.

The movement of fluid by cilia poses an interesting fluid

me-chanical problem which has received considerable attention in the last

quarter century. Two fundamental approaches to the problem have

been advanced. The envelope model has been developed by several

authors including Taylor (1951), Lighthill (1952), Blake (1971a,b,c),

and Brennen (1974); it employs the concept of representing the ciliary

propulsion by a waving material sheet enveloping the tips of the cilia.

(1951) who considered the propulsion of an infinite waving sheet in a viscous, incompressible fluid, neglecting inertia forces. He found that the sheet would be propelled in a direction opposite to the direction of propagation of the wave with a speed ~

(kh)

2

c,

where k == 2rr/A., A. being the wavelength, h is the wave amplitude, and c is the

wave-speed. Taylor's analysis required that kh be much less than unity. The principal limitations of the envelope approach, as discussed in the review by Blake and Sleigh ( 197 4), are due to the impermeability and no-slip conditions imposed on the flow at the envelope sheet (an

as surnption not fully supported by physical observations) and the mathe -matical necessity of a small amplitude analysis (for metachronal

waves, kh is usually of order unity). Furthermore the envelope model is unsuitable for evaluating many important aspects of the problem that arise from the flow within the cilia layer.

In a different approach, Blake ( 1972) introduced the sublayer model for evaluating the flow within as well as outside the cilia layer by regarding each cilium as an individual slender body attached to an

model is caused by neglecting the oscillatory component of the "inter-action velocity". From a physical standpoint, since the velocity of the cilia is typically 2 to 3 times the velocity of the mean flow they pro-duce (Sleigh and Aiello, 1972), one might expect the magnitude of the oscillatory velocities to be considerable. Therefore the velocity that each cilium "sees" at any instant during its beat cycle is likely to be quite different from the parallel mean flow alone.

While the two existing theoretical approaches to ciliary propul-sion have made important contributions to our understanding of a

difficult problem, they also have some drawbacks. The purpose of this investigation is to present a new model for ciliary propulsion intended to rectify the deficiencies in the existing theoretical models. The main features of the model developed in this study are that it allows for finite amplitude movement of cilia and takes into account the oscillatory nature of the flow.

In Section 2 the geometry of the problem is described and the primary ideas are set forth. The central concept of the model is to

replace the discrete forces generated by the cilia ensemble by an equivalent continuum distribution of an unsteady body force, the

2. Description o£ the model

In typical ciliated organisms the thickness of the cilia layer is small compared with the body dimensions. If L is the length of the cilium and 1... is a relevant body length, then it is observed that L/cJ..

<

0. l for rnost ciliated systems (see Tables I and 2). Conse-quently, as a first approximation, the geometry of the model may be taken as a hypothetical planar organism of infinite extent. The cilia are supposed to be distributed in a regular array of spacing a in the X direction and b in the Z direction in a Cartesian coordinate sys-tem fixed in the organism (see Figure 2). Each cilium is assumed to perform the same periodic beat pattern in an XY plane, with a con

-stant phase difference between adjacent rows of cilia, such that the metachronal wave propagates in the positive X direction. This

geom-etry allows both symplectic and antiplectic metachrony, with the exten-sion to other types of metachrony being evident from the discussion.

For any ciliated organism, the basic agency of the motion is the combined effect of the forces exerted by each cilium on the fluid. In view of the difficulty in solving the fluid mechanical problem with a moving boundary so complicated as an ensemble of discrete cilia, an alternative formulation of the problem is developed. As already ex-plained, the central concept of this model is to replace the discrete forces of the cilia ensemble by an equivalent continuum distribution of an unsteady body force field ~(X, Y, t). The vector function !:(X, Y, t)

TABLE 1 EXPERIMENTALLY OBSERVED DATA FROM THE LITERATURE FOR OPALINA RANARUM Quantity Dimension Symbol Values 1 3 2 Body length 1-Lm

t.,

150-500 200-300 220-360 1 3 2 Body width 1-Lm 'w' 100-300 130-2 3 0 160-250 Body thickness

11:

20 l 20 2 20 3 1-Lm Cilium length L 10-20 4 18 5 10-14 8 1-Lm Cilium radius 0. l 6 1-Lm

r 0

Quantity Body length Body width Body thickness Cilium length Cilium radius

Metachrony Wave

y

---A

z

~

II

I

I

I

I

I

X

Fig,

2

Illustration

of

the

coordinate

system

and

regular

i;irray

of

cilia,

The

spacing

in

the

X

direction

is

a,

while

in

the

Z

direction

it

is

For metachronal waves, F is a periodic function of cf>(X, t) ==

kX-wt, where w is the angular frequency of the ciliary beat and

k

=

2rr/A., A. being the metachronal wavelength. Therefore F can be expanded in a Fourier series of the form

00

F(X, Y, t)

=

F (Y)

+

:E Re (F (Y) enicf>],

- -o n=l - n (2. 1)

where Re stands for the real part, i

= .J":T,

and the Fourier

coeffi-cients F (n ~ 1) may be complex functions of Y. Further, since the

-n

cilia are of finite length L, F will vanish above the cilia layer, or equivalently

F(X,Y,t) = 0 for Y

>

YL(kX-wt) (2. 2)

where Y

=

YL(kX-wt) is the thickness of the cilia layer, which depends

on 4> (X, t). The Fourier coefficients F corresponding to the distri-- n

bution (2. 1) will therefore vanish for Y

>

L, or

F (Y)

=

0 for Y

>

L, n ~ 0.

3. Solution of the fluid mechanical_:eroble~

The Navier-Stokes equations for the flow of an incompressible Newtonian fluid in two dimensions may be wrii:ten as

P,x

=

F

+

pvY' U 2 p(U, t

+

UU, X

+

VU, y) ( 3. 1)

P,y ( 3. 2)

where P is the pressure, U and V are the X and Y components

of velocity respectively, F and G are the X and Y components of body force F, p is the density, and v the kinematic viscosity. Here and in the sequel " 11

11 11

' ' X ' ' Y ' and 11

, t" in subscript denote

differ-entiation. The continuity equation, U, X

+

V, y

=

0, will be satisfied

by a stream function 'IF defined by U

=

'IF, y and V

=

-'IF, X. If we

-1 -1 -1 -I

non-dimensionalize all variables using k , w and pvk w as the reference length, time and mass scales respectively, the equations

of motion in non-dimensional variables (lower case) become

( 3. 3)

P,

Y

=

g - \7 2 .I't' ' , X R ("' o't''x'T

+

't'•y't''xx-'t'''I' 'I' .I' x't''I' 'xy ) ( 3. 4)

where 'T

=

wt,

and R

=

w/vk2 is the oscillatory Reynolds number 0

which is taken to be much less than unity as is generally the case

00

f(x, y, T} = f (y}

+

L: Re [f (y}eni(x-T)]

o n=1 n ( 3. 5)

00

g(x, y, T} = g (y) + L: Re[ g (y)eni(x-T)] .

o n=1 n

( 3. 6)

This suggests that we seek solutions for ~ and p of the form

~(x, y, T} = ~ (y}

0

p(x, y, T} = p (y} 0

+

~ Re[~

(y)eni(x-T)]

n=1 n

+

00

L:

n=l

Re[p (y)eni(x-T}] .

n

(3. 7)

( 3. 8)

Substituting (3. 5) - (3. 8) into (3. 3) and (3. 4), and equating the

appro-priate Fourier coefficients, we obtain, after some simplification (see

Appendix A},

00

~o.

yyy

=

-f +R {1/2 L: 0 0

Im[

n~n~

n, y], y} ,

n=1

(3. 9}

00

[

n2~n~n],

y} Po,y = go - R 0 {1/2 L:

n=1

(3. 10)

2 2

~n,yyyy -2n ~ _n,yy + n ~ _n = ;r _n +O(R ), n _o

>

0, ( 3. 11)

P _{n -}-

_!._ [

_ni f _n

+ "·

_{'t'n,yyy- n 't'n,y} 2 "' ]

+

O(R }, _o n

>

0, ( 3. 12)

where Im refers to the imaginary part, an overbar signifies the

complex conjugate, and ;r = nig - f (n

>

0}.

n n n,y

The boundary conditions with respect to the reference frame

condition at the wall, which requires all the Fourier components of velocity to vanish there, that is,

(n

>

0). ( 3. 13)

Second, since we are considering an organism which is propelling itself at a constant finite speed, we require that the mean velocity remain bounded while the 11_AC11 _{components of velocity vanish as} y .-. co. Hence,

4J

(oo)

<

oo,

4J

(co)

=

4J

(co)

=

0

o,y n,y n (n

>

0). ( 3. 14)

We note that the non-dimensional form of the condition (2. 3) is

f _n (y)

=

0 for y

=

kY

>

kL, .

n?

0, and

g (y) _n = 0 for y = kY > kL, n:;?. 0.

( 3. 1 s)

Integrating equations (3. 9) and (3. 10) and applying the boundary conditions (3. 13) and (3. 14) along with the condition (3. 15), we obtain for the mean velocity and pressure

y

fL

y

u = 4; =5 y ₁f (y

1)dy1+ y f (y1)dy1 +R

U

~

Im[S!ll\J

~

dy1]}

o o, y o o o n n, y .

o y n=l o

(3. 16) y

Po= poo

+

5

go(yl)dyl- Ro{t

~

( 3. 17)

o n=l

where p is a constant. Note that as a result of (3. 15), the zeroth co

order terms (in R ) of u and p are constant for y

=

kY

>

kL.

As can be seen from the integrands of ( 3. 16), the chief source

of the velocity at infinity, which corresponds to the speed of propulsion,

is the distribution of the mean x-component of body force within the

cilia layer. The terms of order R come from the coupling of the

0

11_{AC" velocities in the nonlinear inertial terms of the fundamental}

equations. The solution to (3. 11) subject to (3. 13), (3. 14) and (3. 15)

(see Appendix B) is

l\Jn

=

_!_

_4n 2

[a.

_n (y)eny +

~

_n (y)e -ny] + O(R ), _o where (3. 18)

_!.)e-ny1 d

(y-yl - n 1Tn Yl' ( 3. 19)

kL

y .

=

s

(y-yl + !>enyl

kL

1Tndyl-

s

(y+yl+2nyyl+ !)e-nyl 1Tndyl.

0 0 _{(3. 20)}

_ R ["· ni(x-T)]

u -_n e't'_ny 1 e and

The harmonic x and y velocities,

v _n = _-R _{e n1't'ne} [ ·,~, ni(x-T)] _{, n}

>

_{0, are given by}

un

=

Re {

~

[(an y

4n '

+ na )en(y+icp)+ ({3 - n~ )e -n(y-icp)]} + O(R ),

n n, y n o

( 3. 21)

( 3. 22)

where cp

=

x-T. Note from (3. 15) and (3. 19) that a and a vanish

n n,y

for y

=

kY

>

kL, so that the harmonic velocities decay like e -nkY.

The dimensional velocities U and V are obtained by simply

multi-n

n

plying u and v by c

=

w/k,

the wave speed. The solution for the

harmonic pressure coefficients may be determined from (3. 12) and (3. 18) directly.

Finally we observe that for the cilia layer to be self -propelling, the condition of zero m ean longitudinal force acting on the organism can be expressed by

L

=

s

(3. 23)

0

where j..L

=

pv is the dynamic viscosity coefficient; the mean shearing force exerted by the fluid on the wall balances the mean force per unit area of the organism's surface due to the reaction of the fluid to the ciliary body force field. It is of interest to note that the present solu-tion, given by (3. 16) - (3. 20), satisfies condition (3. 23) required for

self-propulsion when the bulk force F of ciliary layer is regarded as known. This is in effect a consequence of the boundary condition imposed at infinity, as Taylor (1951) pointed out. To an observer in a frame fixed with the fluid at infinity (the lab frame), the organism is moving with a constant velocity and hence, by conservation of

4. Force exerted by a cilium

In the preceding section the solution to the fluid mechanical

problem was given as if the body force field F were known a priori.

Since in practice this is not the case, we need to develop a means of

determining F. As an initial step in this regard, we consider the

force exerted on the fluid by a single cilium.

In order to calculate the viscous force exerted by an elongated

cylindrical body moving through a viscous fluid, it is convenient to

utilize the simple approximations of slender -body resistive theory.

If the tangential and normal velocities relative to the primary flow of

a cylindrical body element of length ds are V and V respectively, - s - n

the tangential force exerted by the body element on the viscous fluid

will be dF = C V ds with a similar expression for the normal force - s s-s

dF = C V ds, where C and C are the corresponding resistive

- n n - n s n

force coefficients.

The expressions for the force coefficients have received

con-siderable attention by several authors. The original expressions given

by Gray and Hancock ( 1955) were for an infinitely long circular

cylin-drical filament along which planar waves of lateral displacement are

propagated in an unbounded fluid. Cox ( 1970) developed slightly

modi-fied coefficients to account for other simple shapes. Recent work by

Katz and Blake ( 197 4) have yielded improved coefficients for slender

bodies undergoing small amplitude normal motions near a wall. For

the purposes of the pre sent investigation, however, initially, the force

coefficients developed by Chwang and Wu ( 197 4) for a very thin

ellip-soid of semi-maJ· or axis 1. L

These were found to be

c

_n

=

ln(L/r

0 )

+

t

and

c

=

s

ln(L/ r )

0

1

- 2

( 4. 1)

We may describe the motion of a single cilium with base fixed

at the origin by a parametric position vector

£.

(s, t) specifying the

position of the element ds of the cilium identified by the arc length s

at time t (see Figure 3). The force exerted by the element ds on

the fluid is given by

dF

=

dF

+

dF

- c - s - n

= C V ds

+

C V ds

s - s n - n

where U is the local primary flow velocity,

(4. 2)

( 4. 3)

( 4. 4)

'I = C

I

C , e is a unit

n s - s

vector tangential to the longitudinal curve of the cilium centroid, and

~n. is a unit vector normal to ~ s lying in the plane formed by

s_,

t - U

and e . From elementary calculus it is easily found that e

=

s_,

- s - s s

and

e = [ V- (V

-n - - ( 4. 5)

where V

=

s_,

t - U. Substituting ( 4. 5) in ( 4. 4) yields

( 4. 6)

We are thus left with the task of evaluating F and then converting it

-c

s

= 0

X

[image:26.547.46.457.43.666.2]

5. Determination of the body force field

In order to apply the resistive theory in determining the body

force field F, two difficulties must be overcome. Equation ( 4. 6)

describes the force exerted by an element as a function of its arc

length s. It is convenient to convert this Lagrangian description of

the force to Eulerian coordinates (i.e. to express the force as a

func-tion of X and Y). In addition, the force exerted by an element

depends upon the local fluid velocity, which is initially unknown. Both

of these features suggest

a

numerical iteration approach.

In the following paragraphs the main elements of the numerical

scheme will be described briefly. From (2. l) it follows that

A A

F

0 =

~

s

F (X, Y, t 0 )dX,

0

F =

~

5

F(X Y t ) e -nil/> dX (n

~

l) .

- n A. - ' ' o "

0 _{( 5. 1)}

This implies that if the body force field were known over one

wave-length for some (arbitrary) fixed time t , then all the force coeffi

-o

cients could be determined. In order to accomplish this for a given

ciliary beat pattern, an analytical representation for

5_ (

s, t) can be

determined using a finite Fourier series -least square procedure.

(For the details of this procedure see Blake, 1972 ). Thus the

move-m·ent of a single cilium is given by

N

.s_(s,t) = ~ -oa (s)

+

!: a (s) cos nwt

+

b (s) sinnwt,

n=l-n - n ( 5. 2)

where a (s)

-n

M

= !: A sm

m =l -mn

-n

b (s)

=

M !:

m=l

with A

B being constant vectors, N is the order of.the Fourier series, -mn

and M is the degree of the least squares polynomial expansion used.

Once

.5_

(s, t) is determined, an initial body force field can be

established via a "pigeon-hole11 _{algorithn:J.. Using (5. 2), the cilia are}

numerically distributed over one wavelength at a fixed time t and 0

each cilium is broken up into a finite number of segments. The

Eulerian location of and the force exerted by each segm ent are then

determined and stored (i. e. 11_{pigeon-holed''). The force is evaluated}

using (4. 6) initially with

Q,

the local fluid velocity, equal to zero. The approximation is made that the force exerted by each segment is

located at the midpoint of that segment, so that the force field can be

represented by the expression

1 F(X, Y, t )

=

b

- 0

J

~

j=l

f .o(X-X.) o(Y-Y.),

-J J J ( 5. 3)

where f . is the force exerted by the j -th ciliary segment whose mid--J

point is located at (X., Y .) and J is the total number of segments in J J

one wavelength. The factor 1/b accounts for the averaging in the Z

direction.

Calculations of the body force field can then be carried out by

iteration. Equation (5. 3) is used in (5.1) to determine the Fourier

force coefficients F (n ~ 0). These coefficients, after

non-. -n

dimensionalization, are then used in (3. 16) and (3. 18) to determine a

new primary velocity field !:::!" in which the cilia are assumed to be

immersed. This velocity field is used anew in the pigeon-hole

alga-rithm to determine a new F(X, Y, t ) by computing new f .1 _{s. The}

iterative process is thus repeated until the £ .1 _{s, and hence the}

-J

6. Results for Paramecium and Opalina

Calculations have been carried out on data obtained for Opalina

ranarum and Paramecium multimicronucleatUin. Opalina is a flat disc

shaped ciliated protozoon found in frogs, while Paramecium is a

pro-late spheroidal shaped ciliate commonly found in stagnant water.

Although these two organisms have been observed to have somewhat

three-dimensional beat patterns, they were chosen because of the

availability of the necessary data and for comparison with previous

theoretical work. The raw planar beat patterns were taken from

Sleigh ( 1968) and Tables 3 and 4 give the calculated Fourier series

-least squares coefficients (see (5. 2)) with M

=

N

=

3 for Opalina and

M = N = 4 for Paramecium. Using these coefficients, we present

com-puter plots of the beat patterns in Figure 4. The metachrony of

Opalina is symplectic (Sleigh, 1968), while Paramecium was regarded •

as exhibiting antiplectic metachrony, although it recently has been

reported to be dexioplectic (Machemer, 1972 b).

In order to display the harmonic X-velocity components, it is

convenient to write them in the form

u =

Q cos(ncf>

+

(} ) 1 (n ~ 1), where

n n n ( 4. 1)

Qn

=

c

I

~n·

Y

I

•

and ( 4. 2)

(}

=

_tan-1

[Im(~

)/Re(~

_)]

n n,y n,y ( 4. 3)

For a given value of Y, Qn is the amplitude of the nth harmonic

X-velocity component, while (} is the phase of the motion. This phase

TABLE 3

Fourier series - least squares coefficients for the beat pattern of Opalina drawn by Sleigh ( 1968).

A -ron

n

m 0 1 2 3

1 1. 5170 -0. 1957 0. 0732 0.0838

1. 1290 0. 3079 -0. 1510 -0. 1060

2 -0.0886 -0. 4235 -0. 5698 -0. 327 8 -0. 4102 0. 4428 0.7006 0. 3409

3 0. 1248 0. 3126 0. 3866 0. 2143

0.0298 0. 3903 0. 4920 0. 2007

B --rnn n

m 0 1 2 3

1

...

0. 2801 0. 2157 0.0961

.

...

-0. 2855 -0. 1250 0.0225 [image:31.546.47.481.53.549.2]

2

....

..

-0. 5157 -0. 3445 -0.0548

...

0.7350 0. 2241 -0. 3343

3

.

...

0. 2163 0. 110 5 -0. 0197

...

-0. 2740 0.0386 0. 3104

TABLE 4

Fourier series - least squares coefficients for the beat

pattern of Paramecium drawn by Sleigh ( 1968). A

-mn

n

m 0 1 2 3 4

1 -1. 2580 0. 1458 -0. 3606 0.0663 0.0534

I. 5270 0. 179 5 -0. 1878 -0.0074 -0.0063

2 I. 3528 -0. 1122 0. 0466 -0.2385 -0. 277 4 -0.4827 -0. 7740 1. 0896 -0. 1997 -0. 5204

3 -0.0920 -1. 9370 0. 8160 0. 3980 0. 4446

o.

6804 I. 4682 1. 7339 0. 2671 0. 1937

4 0.0928 I. 4216 -0. 7118 -0. 2339 -0. 2340

-0. 4941 -0. 6901 0. 8636 -0. 1165 -0. 0924

B

-mn n

m 0 1 2 3 4

[image:32.546.52.480.136.627.2]

1

...

0.0565 0. 3987 -0. 0502 0.0851

...

0. 1734 -0.0863 0. 0406 -0.0085

2

...

2. 5308 -2. 5708 0. 1122 -0. 3483

...

-0. 2149 0.7584 -0.0888 0. 0 421

3

...

3.2568 4. 2679 -0.2352 0. 5374

...

-0.2868 0. 9381 0. 1863 -0. 0492

4

...

I. 0828 -2. 1502

o.

1243 -0. 2599

...

0. 3610 0. 3697 -0. 0867 0.0136

Note: The length of the cilium has been normalized to 1. Upper and

lower coefficients in each box correspond to X and y component of

is such that U ts at a maximum and in the same direction as the

n

mean flow when cf>

=

(21 1r - (} )/n, while U is at a maximum and

n n

opposes the mean flow when c/J

= [

(21 -1) 1r - (} ] /n, where 1 is an n

integer.

In performing the computations, besides the basic beat pattern,

it was necessary to specify the three-dimensionless parameters ka,

kb, and kL. The values of the parameters were chosen to be

consis-tent with the consensus of experimental observations (see Tables 1 and

2). The values used for ka and kb were 0. 63 and 0. 13 for Opalina

and 0. 80 and l. 60 for Paramecium. Two values of the amplitude

parameters kL, to which the computations were the most sensitive,

were used for each organism, these being l. 25 and 2. 50 for Opalina

and 3. 00 and 6. 00 for Paramecium. Because of the apparent rapid

decay of the higher harmonics and for computational efficiency, only

the first harmonic was retained in computing the local velocity field

U in three cases, however, the second harmonic was included for

Paramecium with kL = 3. 00.

Figures 5 and

6

show the velocity profiles for U , the mean

0

fluid velocity parallel to the wall, for Opalina and Paramecium

re-spectively. Also plotted are the profiles given by Blake ( 1972)

(dashed curve). The velocities at infinity, U00/c, which correspond

to the velocities of propulsion, are seen to agree quite well with the

observed values of 0. 5 - 1. 5 for Opalina and 1 - 4 for Paramecium

(Brennen, 1974). Blake's calculations were made with the quantity

(ka) (kb) equal to 0. 04 for Opalina and 0. 0025 for Paramecium (both

y

T

co

~r---,.--.---~.---r---~

::f'

.

-kl= 1.25

kl=2.00

kl =2.50

N •

-0

.

-CD

.

0

c.o

•

0

::f'

.

0

N

.

0

0

.

0~---~~---~---~---~

-0.5

0.0

0.5

1.0

1.5

Uo/c

[image:35.549.28.474.54.655.2]

y

L

( 0

.

-

I

I

I

:::1'

.

I

-

I

I

I

N

I

.

-

I

kL = 0.5

I

0

.

-kL=3

CD

.

0

(0

.

0

. :::1'

•

0

N

.

0

0

.

OL---~---L---~---L---J---~

-1.0

o.o

1.0

2.0

3.0

q.o

s.o

Uo/C

Fig; 6 Mean velocity profiles for Paramecium with ka

=

0. 80

and kb

=

1. 60. Dashed curve is from Blake (1972) for

While Blake 1 _{s re suits indicate a slight backflow for only the antiplectic}

case, the present study indicates a backflow in the lower half of the cilia layer for both Opalina and Paramecium (for kL

=

6).

Figure 7 shows the amplitude of the first harmonic velocity

component U I for Opalina. The largest oscillatory velocities appear to occur at Y /L

=

6

and are more than twice the mean flow in

magni-tude. In Figure 8 the amplitudes of U I for kL equal 3 and 6 and

u

₂ for . kL equal 3 are shown for Paramecium. The largest oscillatory

velocities occur in the upper part of the cilia layer and are about the same magnitude as the mean flow. Note that for kL

=

3 the amplitude

of

u

2 is half that of U I.

The corresponding () 's for Figures 7 and 8 are shown in

n

Figures 9 and IO respectively. The origin and the value of t (see

0

section 5) were chosen so that

¢

=

0 corresponds to a metachronal wave peak under which the cilia are in the effective stroke. This is illustrated in Figures 4b, d which can be regarded as snapshots of the cilia at t

=

t .

0 For Opalina at Y

I

L

=

0. 5 and cp

=

0,

=

cos (8

1) equals approximately 1 for kL

=

I. 25 and 0 for kL

=

2. 5. This means that U

1 is somewhat in the same direction as the mean flow in the vicinity of the cilia performing the effective stroke.

How-ever, above Y /L

=

0. 5 U I progressively tends to oppose the mean

flow. For ParameciUin. however, at Y/L = 0. 5 and

¢

= 0, UI/Q 1

equals -I (approximately) for both kL values. Thus, in this case,

U

1 tends to oppose the mean flow in the vicinity of the effective stroke.

It is also seen that U

2 (for kL = 3) is in the same direction as the

y

L

co

~r---~---.---1

::f'

•

-N

.

-0

..

-

kL=1.25

CD

..

0

CD

.

0

::f'

•

0

N

...

0

0

..

0~---~---L---~---~~---~---~

0.0

0.5

1.0

1.5

2.0

2.5

3.0

a,lc

[image:38.552.55.475.42.637.2]

co

_:r..---·

::fl

•

-N

.

-c

•

-y

T

co

.

0

to

.

0

::fl

..

0

N

..

0

c

.

0~---~~---~---~---~---~---~

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0

₀

/c

Fig. 8 Amplitude profiles for nthharmonic X-velocity components

y

T

co

~r---r---,---,

::fO

•

-N

•

-0

•

-CD

•

0

co

.

0

::ft

.

0

N

.

0

0

•

0~---~---~--~---~---~

2

1T

*

27T

5217"

e,

Fig. 9 Variation of 8 l, the phase of the first harmonic X -velocity

[image:40.541.45.483.44.652.2]

y

L

CD

~~---..---~-.---,

::1'

.

-N

.

-0

•

-CD

.

0

co

•

0

::1' •

0

N

.

0

0

.

0

7r

2

n =I ,

kL= 6

Bn

[image:41.540.45.465.49.663.2]

7. Energy expenditure by a cilium

Once the final velocity field

:Q

has been determined, the rate of energy expenditure by an individual cilium can be calculated. The rate of working of a single cilium, E ,

.

is the sum of the rates of

c

working of the cilium elements. Therefore we may write L

Ec(t)

=

S

(.g_,t(s, t ) -

u

J

dF - c

dS (

s, t)ds, (7. 1)

0

where U is regarded as a function of s and t. This integral was carried out numerically for Opalina and Paramecium using the final f . 1 _{s (see section 5) obtained in the computations of the previous section.} -J

In specifying

Q,

in addition to the mean flow, both the first and second harmonics were retained.

Figures 11 and 12 show the power expended by a cilium over one beat cycle for Paramecium and Opalina respectively. The power,

•

E , on the vertical axis has been non-dimensionalized by the quantity c

f.Lc2 /k. The rate of working for Paramecium is greatest during the effective stroke, while Opalina has two peaks of power output, one during the effective stroke and one during the latter part of the return stroke. Note that the power expenditure is very sensitive to the

amplitude parameter kL; the upper curve in each figure, which corresponds to the higher kL value, has been scaled by a factor of 10 -I.

80

I

T

kL

ko

kb

----XI

o-1

28

6

0.80

1.60

7

3

0.80

1.60

60

---150

0.5

-BLAKE

(

1972)

.

E

..-\

-I

12

(~2)

I

\

k

\

401--/

\

I

I

....

----,..,

/

...

_,/

...

"

7T

.

14

EFFECTIVE--~---RETURN

Fig. 11 The power expended by a cilium over one beat cycle for Paramecium,

•

E

(if)

4

I -I

----X

10

3J-

---2

"

I

\

''

I

\

/

-I

\

T

79

19

150

/

\

/ \.

-,

\

0

'-

r-EFFECTIVE

kl

ko

kb

2.50

0.63

0.13

1.25

0.63

0.13

0.50

-BLAKE

( 1972)

NHARM

=

0

Tr _{_I} __

-.

(\

I

\

I

\-,

I

\

I

/""'\

\

I~

\

I

I

It is seen that when the oscillatory velocities are neglected, the energy expenditure over one beat cycle is substantially less·than that obtained with the oscillatory components included. The corresponding curves obtained from the analysis of Blake ( 1972) are also indicated on the figures. In the present notation, the parameter T used by Blake is

given by C (kL)2

I[

f.L(ka)(kb)]. Because of the discrepancies in the s

values of T and kL used, a comparison of the results is difficult. Finally it is of interest to estimate the actual work done,

w,

c by a cilium against viscosity over one beat cycle, which is given by the area under the curves of Figures 11 and 12 times the quantity f.Lc2 /k. For Opalina this quantity is about 35 x 10 -lO ergs

I

sec, while for Paramecium it is about 92 x 10-lO ergs/sec. Therefore it is found that for Opalina with kL

=

2. 5, W

=

1 x 10-8 ergs and for Paramecium

c -8

with kL = 3, W

=

0. 4 x 10 ergs. These values are in agreement c

with an estimate of the work done by a component cilium of the com-pound abfrontal cilium of Mytilus obtained from the experimental work of Baba and Hiramoto (1970). Hiramoto (1974) estimates that the work

-8 -1

done by a single component cilium is 1 x 10 erg beat Further-more, Sleigh and Holwill ( 1969), using a biochemical approach based upon the number of ATP molecules de-phosphorylated per beat,

8. Conclusions and discussion

The concept of replacing the discrete cilia ensemble mathe-matically by a layer of body force leads to a reasonably successful

model for ciliary propulsion. The principal advantages of this approach are that it allows for the large amplitude movement of cilia and takes into account the oscillatory aspects of the flow.

When applied to Opalina and Pararn.ecium, a good agreement is found between theoretically predicted and experimentally observed velocities of propulsion. The amplitude of the oscillatory velocities 1s found to be of the same order of magnitude as the mean flow within the layer and to decay exponentially outside the cilia layer.

The energy expenditure by a single ciliurn over one beat cycle is found to be very sensitive to the amplitude of the ciliary movement and to be significantly greater than previous theoretical estimates which neglected the oscillatory components of the flow field. The work done by an individual cilium against viscosity during one beat cycle is found to be consistent with experimental and biochemical estimates.

The principal limitation of the model is due to the geometry employed. The main objection is that an infinite flat sheet model is limited in its capacity to assess the effect of the finite

three-dimensional shape of the organism. While this criticism is valid, because of the mathematical complexities involved, it seems likely that geometrically simplified models will contribute most to our

A second limitation of the present approach is with regard to

the approximations of the slender-body resistive theory used. The

problem of calculating the force exerted by a slender body moving in

a given primary flow field in close proximity to many neighboring

bodies and in the vicinity of a wall is an enormous one. It is felt that

the present resistive theory approach provides a reasonable alternative

to the task of rigorously considering these complications. The effect

of the wall is taken into account by imposing the no-slip condition upon

the flow, while the influence of the neighboring cilia elements is

effectively represented in the local primary flow field. While the

resistive coefficients employed, C and C , were developed for a

n s

slender body immersed in a uniform primary flow, it is reasonable to

expect that these coefficients would be useful approximations for

non-uniform primary flows. In this regard the work by Cox ( 1971) on a

slender body immersed in a primary shear flow indicates that more

Appendix A

Derivation of equations (3. 9) - (3. 12)

Equations (3. 9) - (3. 12) are obtained from the substitution of

the Fourier expansions for ~' p, f and g given by ( 3. 5) - ( 3. 8) into the equations of motion (3. 3) and (3. 4). We consider the linear and

nonlinear terms separately. The linear terms needed are

f

=

g =

p,x =

p,y

=

~'xT

=

~'y'T

=

'V2~

'y =

'V2~

'x

=

00

f

+

~

0·

go

+

Po,y

+

~0,

yyy

+

n=l

00 ~

n=l

00 ~ n=l 00 ~ n=l 00 ~ n=l 00 ~ n=l 00 ~

n=l

00

L;

n=l

where r.p = x-T,

nir.p Re(g e ),

n

Re(nip enir.p), n

nir.p Re(p e ),

n,y

Re(ni~ enir.p),

n,y

n 2

~

) e nir.p ] ,

Re[

~

n, yyy _n,y and

3.~ ) nir.p]

Re[ (ni~ - n 1 e

n,yy n

(A. 1)

(A. 2)

(A. 3)

(A. 4)

(A. 5)

(A. 6)

(A. 7)

Before considering the nonlinear terms, we note that if z

1 and z

2 are two complex numbers and m and n are integers, then

(A. 9)

Therefore it is easily seen that

00

4J,y4J'xy _n=~ Re(ni~ 4J )

1 n,yn,y +ACterms, (A. 10)

00

lfi,x4J'yy = 2 1 ~ Re (ni4J 4J ) + AC terms , (A. 11)

n=l n n,yy

00

2

-1 _{+ AC terms , (A. 12)}

4J,y4J'xx = 2 ~ Re(n 4J ~ ) n=l n,y n

00

2

-4J,x4J'xy = 2 1 ~ Re(n ~ 4J ) + AC terms. (A. 13) n=1 n n,y

Using (A. 10)- (A. 13), we easily determine the needed nonlinear terms

to be

00

4J, y~' xy -

lf!,

x4J' yy

=

!

;= 1 Re [ ni(4Jn, y4in, y + 4Jn4Jn, yy)]

+ AC terms

(A. 14)

00

=

-!

~ Re[ni(4J 4J ) ] + AC terms n=1 n n,y ,y

00 - l

- 2 ~ Im[ n(4J 4J ) ] + AC terms,

= ~ 00 ~ Re(n 2

(l\J

-l(! + l(! l(! - )] + AC terms n=l n,y n n n,y

=

00

~

L: Re( n2(l(J

~

) ] + AC terms n=l n n 'y

= (A. IS)

Upon substituting (A. I) - (A. 8), (A. I4). and (A. IS) into (3. 3) and ( 3. 4) and collecting terms, we find that

0

0

00

= fo +

l\J

o, YYY - Ro(} :=I

Im(ruvn~n,

Y), Y]

+

~

Re {( (£ -nip + l(! - n2l(J )+ R (nil(! )]eniq7} n= I n n n, yyy n, y o n, y

+ R (AC terms), and

0 (A. I6)

oo R { ( ( . 3 ) R (n2"' ) ] eniqJ}

+L: e g - p -ml(J +nil(!+ 't"

n= ₁ n n, y n, yy n o n

+ R (AC terms) .

Since each Fourier coefficient in (A. 16) and (A. 17) must vanish,

00

~o,yyy = - f +R[~ ~

0 0

n=l

(A. 18)

(A. 19)

= -

1 [ f

+

.t. -

n

2

o~

.

]

+

O(R ) , n

>

0, and

ni n 't'n,. yyy 't'n, y

o

(A. 20)

=

g -

ni~

+

n 3 iljJ

+

0 ( R ) , n

>

0 .

n n, yy n o (A. 21)

Equations (A. 18), (A. 19), and (A. 20) correspond to (3. 9), (3. 10), and (3. 12) respectively. Equation (3. 11) is obtained from (A. 21) by

~ndixB

Derivation of solution to (3. 11) subject to (3. 13), (3. 14), and (3. 15). To order R equation ( 3. 11) may be written as

0

where D

=

_dy d 1

or upon factoring the operator, we have

2

2

(D - n) (D

+

n) tV

=

'IT_ •

n n

2

If we let CI?

=

(D + n) tV , then (B. 2) may be written as

n

CI?'yy

2

2n <ll, + n CI? = 'TT •

y n

(B. 1)

(B. 2)

(B. 3)

The solution to this second order, linear, inhomogeneous equation may be written down as

y

(D+n)2tVn =(Any+ Bn)eny +

s

(y-yl)en(y-yl)'TTn(yl)dyl, (B. 4)

0

where A and B are arbitrary constants. Similarly, upon

re-n n

placing n with -n we have

y

(D-n)2tVn= (Cny + Dn)e -ny +

5

(y-yl)e -n(y-yl) 1Tn(yl)dyl, 0

(B. 5)

where C and D are arbitrary constants. Subtracting (B. 5) from

n n

(B. 4) we obtain

0

Similarly, adding (B. 4) and (B. 5) we have

ny -ny

= (A y

+

B )e

+ (

C y

+

D ) e

n n n n

(B. 7)

0

Elimination of

4J

from (B. 7). using (B. 6) yields the general

solu-n,yy

tion to (B. 1) which may be written as

4J

_n

=

y

1 I

s

I -ny ] ny

-2

{((A

(y- -)+ B

J

+

(y-yi- -)e I rr (y1)dy1 e

4n n n n n n

0

(B. 8) y

+ [[Cn(y+ !)+Dn]+

s

(y-yl+ !)enylrrn(yl)dyl]e-ny } , n

>

0.

0

The constants A , B , C , and D are now determined by

n n n n

imposing the boundary conditions (3. 13) and (3. 14):

4J

(

0 )

=

0 =I D

= B

,

n,y n n (B. 9)

4J

(0)

=

0

=>

C

=

A - 2nB ,

n n n n (B. 1 0)

(B. 11)

Note that as a result of ( 3. 15), rr vanishes identically for y

>

kL, n

so that the upper limits of integration in (B. 11) and (B. 12) need only be kL. Also from (B. 6) it is seen that (B. 11) and (B. 12) are compat-ible with the condition

lfi

(oo) = 0.

n,y Upon substituting (B. 9) - (B. 12) into (B. 8), we obtain that to order R,

0

l\Jn

=

_4n2 1

y

{[s

(y-y1 kL

1 -ny ny

- -)e 1 rr (y )dy ] e

n n 1 1

y kL

+

[s

(y-yl

+~)eny1

iTn(y1)dyl- s(y+yl+2nyyl+! )e-nyl

0 0

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Winet, H. (1973). Wall drag on free-moving ciliated micro-organisms,

J. exp. Bioi.

2.2_,

7 53-7 66.

Wu, T. Y. (1973). Fluid mechanics of ciliary propulsion. Proceedings

of the Tenth Anniversary Meeting of the Society of Engineering

Science. Publisher: Society of Engineering Science, Raleigh,

PART II

1. Introduction

The manner in which various micro-organisms propel

them-selves, while captivating micro-biologists for centuries, has only

recently attracted the attention of hydrodynamicists. Although

micro-organisms are capable of moving in many fascinating ways, there are

two predominant types of locomotion which taxonomists have long used

as a basis for classification. Flagellates consist of a head and one or

more long slender motile organelles called flagella. In almost all

cases they move by propagating a wave or bend along their flagellum

in a direction opposite to that of the overall mean motion. Ciliates, on

the other hand, consist of a body that is covered by a great number of

hair-like organelles called cilia. They move as a result of the

con-certed action produced by the beating of each individual cilium.

Since micro -organisms are minute in size, the Reynolds

num-ber based upon a certain ch;,tracteristic dimension of the body 1 and

the speed of propulsion U is very small, i. e. R

=

U.t/v

<<

1,

e v

being the kinematic viscosity coefficient of the liquid medium. For

the motion of most protozoa, the Reynolds number is of the order of

-2

10 or less. Hence the predominant forces acting on

micro-organisms are viscous forces, the inertia forces being negligible.

This fact prompted Taylor (1951) to pose the following question:

"How can a body propel itself when the inertia

forces, which are the essential element in

self-propuslion of all large living or mechanical

bodies, are small compared with the forces due

In attempting to answer Taylor1 _{s question, most fluid mechanical}

research on cell motility so far has been on flagellated

micro-organisms. This is probably because they are the easier system to

model theoretically. The inert head creates a drag while the beating

flagellum provides a thrust, and the condition that the thrust balance

the drag yields a relationship between the speed of propulsion and the

other fluid dynamical parameters of the motion. On the other hand,

as the thrust and drag are inseparable for the momentum consideration

of a ciliated micro-organism, hydrodynamical models proposed for

dealing with the phenomena are necessarily more complex than those

for flagellates. In contrast to the fluid mechanical research, however,

most biological research on protozoa, in which motility is often used

as an assay of internal cell functions, -has been devoted to the ciliates.

This is principally because, in comparison to the flagellates, they are

easier to obtain and maintain in the laboratory.

Primarily due to the great complexity of ciliary movement,

the existing theoretical models proposed for predicting the motion of

ciliates have been noted to contain considerable shortcomings in

ade-quately representing the physical phenomena. Several authors,

including Blake (197lb, 1972), Brennen (1974), and Keller, Wu, and

Brennen ( 197 5) have considered geometrically simplified models

involving planar or cylindrical surfaces of infinite extent. These

infinite models have made important contributions to our understanding

of several aspects of the problem. However, while these models can

to the radius of curvatu:r-e of the cell surface is small, they are limited in their capacity to assess the effect of the finite three-dimensional

shape of the organism upon its locon'lotion.

There have also been several theoretical investigations of

ciliary locomotion using a spherical geometry. Lighthill ( 1952}, Blake (197la) and Brennen (1974) have considered the propulsion of spheri-cally shaped micro-organisms using an envelope model based upon replacing the organism by a material envelope undergoing small ampli-tude undulations about a spherical shape. In each of these investiga-tions, the assurnptions were made that the fluid adheres to a material

envelope covering the cilia (the no -slip condition) and that the envelope propagates small amplitude waves. While this approach has also

illuminated certain important features of the phenomena, as pointed out by Blake and Sleigh (1974), the no-slip condition imposed on the flow at the envelope surface, and the small amplitude of the waves are

assumptions not fully supported by physical observations.

To overcome these limitations, Blake (197 3) considered a

different approach to a finite model for a spherical ciliated micro-organism. In this approach two regions were considered: one being the cilia layer and the other the exterior flow field. The velocity field for the cilia layer was determined using an infinite flat plate model

(c£. Blake, 1972), while the velocity field for the external region was

taken to be the known solution for a translating sphere with both radial and azimuthal surface velocities (cf. Blake, 197la). The two velocity fields were then matched at a spherical control surface covering the

into the exterior field and the propulsion velocity could be determined.

While Blake's finite model sheds considerable light for spherically

shaped organisms, its validity for the large number of non-spherical

shapes found in nature is not entirely clear.

In order to assess the effect of geometrical shape upon the

locomotion of a ciliated micro -organism, a theoretical analysis is

carried out in the present investigation to consider a prolate spheroidal

shaped body of arbitrary eccentricity. The central concept of the

model is to represent the micro-organism by a porous prolate

sphe-roidal control surface upon which certain boundary conditions are

prescribed on the tangential and normal fluid velocities. The principal advantage of this approach is that it allows the overall effect of the

cilia to be represented by prescribed boundary conditions on a simple

geometrical surface. Furthermore, since it has been shown that the

oscillatory velocities created by the cilia decay rapidly with distance

from the cell wall (Brennen, 1974; Keller, et. al., 1975), only the mean fluid velocities need be considered.

The development of the model proceeds in a manner aimed at

understanding the mechanism of propulsion and the effect of the shape

of the body on its locomotion. The geometry and boundary condifions

are set forth explicitly in Section 2. As the viscous forces are

pre-dominant in comparison with the inertial forces at the low Reynolds

numbers characteristic of micro-organism movement, the hydrody-namic e·quations of motion may be approximated by Stokes equations.

In Section 3 these equations are solved subject to the appropriate ·

acting on a freely swimming organism must vanish then yields a

relationship between the speed of propulsion, the parameters that

characterize the boundary conditions, and the shape of the organism.

Using this relationship in Section 5, we consider the energy expenditure

by a ciliated micro-organism. These considerations lead in Section 6

to an estimate of the effect of shape upon the power required for the

motility of a ciliate. In order to visualize the flow, an expression is

obtained in Section 7 for the stream function of the flow field and the

streamlines are determined and discussed for several cases. Finally,

in Section 8 several experimental streak photographs of small neutrally

bouyant spheres suspended in the fluid around actual micro-organisms

are presented. In order to exhibit the drastic difference between the

case of a self-propelling micro-organism and that of a dead specimen

sedimenting under gravity, streak photographs have been obtained for

both types of flow and are compared with their respective theoretically

2. Geometry and boundary conditions

Although ciliates have a wide variety of body shapes, a great

number of them are approximately axisymmetric and can be suitably

represented by a prolate spheroid. Some common examples include

Tetrahymena pyriformis, Spirostomum ambiguum, and Paramecium

multimicronucleatum (see figure 1 ). The equation describing the

prolate spheroidal control surface S which represents the organism is

2 2

= X

+

y ' a;? b), ( 1)

where a is the semi-major axis and b is the semi-minor axis. The

focal length 2c and eccentricity e are related by

c =

=

ea (O~e<l). ( 2)

The organism 1s assumed to be translating with respect to the

laboratory frame with velocity U

=

Ue where U

>

0, and e is a

-z

-z

unit vector pointing in the plus z direction. The velocity field with

respect to the laboratory frame will be denoted by

~J.

The velocity

field ~ with respect to a coordinate system, fixed with respect to the

organism (the body frame), with origin coinciding with and axes

1

parallel to the laboratory frame is related to u by

1.

u

=

u

+

u

(3)

In this formulation of the problem, the effect of the cilia on the

fluid is represented by certain pr